solve each equation if 0 < x < 2pi. answer must be in radians

1. 4 cos x - 2 = 0
2. 3 sec^2 - 4 = 0

1. 4cosx = 2

cosx = 2/4 = 1/2
I know cos π/6 = 1/2 , so x = π/6
but the cosine is positive in quadrants I or IV
so x is also 2π- π/6 = 11π/6

#2
sec^2 = 4/3
1/cos^2 x = 4/3
cos^2 x = 3/4
cos x = ± √3/2
so x could in any of the 4 quadrants
you should know the angle whose cosine is √3/2

you will have 4 answers, let me know what you got.

1.

cos x = 1/2
x = 60 or x = -60 which is 300 in degrees
or
x = pi/3 or x = 2pi-pi/3 = 5 pi/3 in radians

1. 4cos(x)-2=0

cos(x)=1/2
x=acos(x)=1/2 between 0 and 2π
or π/3 or 5π/3 because cosine is positive in the first and fourth quadrants.

2. 3sec²(x)-4=0
3sec²(x)=4
cos²(x)=3/4
cos(x)=±(√3)/2
for
cos(x)=+(√3)/2
x=π/6, 11π/6
cos(x)=-(√3)/2
x=5π/6, 7%pi;/6

Back substitute all solutions in original equations to check the answers.

Oh my!

I know that cos π/3 = 1/2 and cos π/6 = √3/2
(What is the matter with me ? )

I guess we all started at the same time, and I am the slowest of us all! :)

To solve these equations, we need to find the values of x that satisfy the equation within the given range of 0 to 2pi (excluding endpoints).

1. 4 cos x - 2 = 0:
First, let's isolate the cosine term:
4 cos x = 2
Now, divide both sides by 4 to get:
cos x = 1/2
To find the values of x, we need to determine the inverse cosine (also known as arccos) of 1/2.
Using a calculator or reference table, we find that the angle whose cosine is 1/2 is π/3 and (2π/3) in the range of 0 to 2pi.

Therefore, the solutions to the equation 4 cos x - 2 = 0 in radians are x = π/3 and x = 2π/3.

2. 3 sec^2 x - 4 = 0:
First, let's isolate the secant term:
3 sec^2 x = 4
Now, divide both sides by 3 to get:
sec^2 x = 4/3
To find the values of x, we need to determine the inverse secant (also known as arcsec) of 4/3.
However, it is important to note that the range of secant function is restricted to (−∞,−1] and [1,+∞), so there are no solutions to this equation within the given range of 0 to 2π.

Therefore, the equation 3 sec^2 x - 4 = 0 has no solutions in the range of 0 to 2pi.